Determine how many solutions exist for the system of equations. ${-9x+3y = -30}$ ${y = -x+6}$
Solution: Convert both equations to slope-intercept form: ${-9x+3y = -30}$ $-9x{+9x} + 3y = -30{+9x}$ $3y = -30+9x$ $y = -10+3x$ ${y = 3x-10}$ ${y = -x+6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x-10}$ ${y = -x+6}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.